There is a view that faith and belief are not the same, for the former is to have assurance about a thing, while the latter can include doubt. I don't think this is really about a difference between the words "faith" and "belief," because both words can have the same meaning. It seems the issue is whether there is a faith or belief that is defined as assurance and certainty, and a faith or belief that includes doubt.
The book of Hebrews describes faith as "the assurance of things hoped for, the conviction of things not seen." (Hebrews 11:1, New American Standard) But some want to argue that faith can include doubt, that faith is not necessarily assurance. There can be a faith that is assurance concerning a thing, while there is a faith that includes doubt concerning a thing.
The faith that includes doubt could be in which a person may act with a certain degree of trust but has doubts about what will happen. For example, you could trust a friend due to his known character but be uncertain that he will be able to fulfill what he claims.
I see two things in view, as in the example just given, of which faith and doubt can latch onto. It's not that I am having faith and doubt in the same thing at the same time. There can be two objects of faith, one is the character and integrity of the individual that makes a promise or claim, and the other is in what they can promise or claim. Humans can fail, no matter how trustworthy their character. If they were God, you could believe their character and their claim, because you could not only believe in their character, but you could believe they would not fail in their claim.
If belief is certainty, does that mean you have to have absolute knowledge concerning a thing? I don't believe you do. To be certain just means you are certain, it doesn't mean you have to have absolute knowledge on a metaphysical level (like God can have). Does certainty mean that your belief has to be true? I don't believe it does, because you can believe in something that is not true, yet your belief is still belief. Certainty is not uncertainty.
Is uncertainty unbelief? It may not be a stubborn rejection of something, but uncertainty is not certainty: it involves doubt, whether it be a mild doubt or a strong one. Having doubt is not necessarily bad; it just means you are not certain: you are not persuaded. We can see in Scripture examples where faith and doubt are in contrast, such as when Peter begins to walk on water at the command of Christ, and yet begins to sink when he is frightened. Jesus asked him why he doubted. Peter started out with certainty, but then uncertainty took over. Jesus said to Peter, "O you of little faith, why did you doubt?" Matthew 14:31 What was the "little faith?" Maybe it recognizes that Peter believed Jesus could enable him to walk on water, but when he saw the waves, he became uncertain that he could walk on the water. Peter limited the ability of Christ. This is in contrast to the "great faith" of the Centurian in Matthew 8:5-13.
Jesus said that the Centurian had "great faith." I believe the circumstance in which he believed revealed it as "great," not that there are different quantities of faith. In Matthew 8:5-10, Jesus says to the Centurian that he will come to his house to heal the one in need, but the man says that Jesus does not need to go to his house but only speak the word from where he is at. His faith is "great" because he believed Jesus could heal by simply speaking the word where he was at without coming to his house. Others would expect Jesus to go where the healing was needed, but this man did not. That was what made his faith great.
There was the situation where Jesus said, "All things are possible to him who believes.” Immediately the boy’s father cried out and said, “I do believe; help my unbelief.” Mark 9:23-24 This is used as an example of one having belief and doubt at the same time. We don't know exactly what the man meant when he responded this way. It could be his way of saying he wanted to believe, but he had doubts. Or it could be that the man believed one thing but had doubts about another. It would be like saying that you believe God is able to do such and such, but you don't know that he will. I believe God can heal me, but I don't know that he will, or I doubt that he will. I could believe that he will, if he gave me special revelation, but typically, that revelation doesn't come. So maybe the man believed Jesus could cast the demon out of his son, but he didn't know if he would. Or maybe it was just his way of saying he wanted to believe Jesus could do it, but still wasn't sure he could.
If a person believes in a thing (called "X"), but there is a view (called "Y") that would refute X, and that person does not believe in Y, but he acknowledges that Y is plausible, does that mean he doesn't really believe in X, or does he believe in X but with doubt (maybe a weak doubt)? Is this an example of faith with doubt? To say that Y is plausible, when Y refutes X, would seem to suggest some doubt about X. Maybe it's just exercising some humility about the subject matter of which X and Y involve.
Sometimes we believe in something that has arguments against it--that have some validity, but you are not persuaded by them, but you recognize them. Sometimes we embrace a view because it makes the most sense and/or the majority hold to it, and yet there are other views that have arguments that seem plausible or valid, which you may not have the ability to refute, and yet you go with a different view. It may not be doubt concerning those opposing arguments, but a recognition that they are valid or difficult to refute, but you take a different view. And why do you take that different view? Hopefully it is due to honest persuasion by the evidence and not just a stubborn rebellion.
With the options of God existing (X) and not existing (Y), the former X seems more likely to you that Y. You are convinced that the X option is the better one of the two, though you have doubts about X. Being convinced X is the better option, you go with it and live accordingly. There's two objects here for faith: one is which is the better option; the second is which is true. You are certain of one, but have doubts about the other. You live according to X because you are persuaded it's the better option, but you have doubts that it is true. I think this could be problematic in your walk with God, because of what Hebrews 11 says. It seems like if I am uncertain about the existence of God, then everything else kind of hangs in doubt that is connected. Hebrews 11:6 says, "But without faith it is impossible to please Him, for he who comes to God must believe that He is, and that He is a rewarder of those who diligently seek Him."
If I doubt the divine authority of Scripture, then I'm more likely to doubt it's claims and promises. But even here, it could be like the belief in God example: I am certain that going with the divine authority of Scripture view is the better option, though I am uncertain that it is divinely authoritative. But that uncertainty will probably cause you problems, even if you believe it is better to live your life as if scripture is divinely authoritative, because when difficulty comes, you can have a reason to disobey its instruction.
If says they have doubts about everything, that seems rather exaggerated. Would you go to work every day, if you doubted you would be paid? Faith is not about perfect knowledge but whether you are persuaded. It's not doubt if I turn out to be wrong in my belief, because I can be convinced and wrong; it just means I was fooled.
If a person has doubt about a thing, then they are not yet persuaded --understanding persuasion as belief. If you are not yet certain, then it seems you do not believe it. If you do not have assurance about a thing, then you do not yet believe in such a thing. You may have to make a choice between two things of which both you are not sure, so you go with the thing that seems to have the better argument(s) or has the most support, but there's still uncertainty. A person may want to believe something, and they may be close to believing it, but they still are not certain about it: they lack assurance and conviction about it.
The author of Hebrews describes faith as assurance and conviction about things God has promised, and he gives many examples from the Old Testament of how it looks when one lives by faith. That kind of faith believes in both the character of God and God's ability to fulfill what he promised. Those are 2 things which are connected. To not believe in both would be to have faith and doubt, but ultimately, in that case, it would be to question the character of God, because God is not human, and he cannot fail. Either you have assurance and conviction in what God promises or you are still unbelieving, whatever the reason or however close to believing you are; you are still not persuaded. Faith is assurance and conviction.